Detailed revision

Vectors algebra

Introduction.

Vectors represent one of the most important mathematical systems, which is used to handle certain types

of problems in Geometry, Mechanics and other branches of Applied Mathematics, Physics and Engineering.

Scalar and vector quantities : Physical quantities are divided into two categories scalar quantities and

vector quantities. Those quantities which have only magnitude and which are not related to any fixed direction

in space are called scalar quantities, or briefly scalars. Examples of scalars are mass, volume, density, work,

temperature etc.

A scalar quantity is represented by a real number along with a suitable unit.

Second kind of quantities are those which have both magnitude and direction. Such quantities are called

vectors. Displacement, velocity, acceleration, momentum, weight, force etc. are examples of vector quantities.

Representation of Vectors.

Geometrically a vector is represented by a line segment. For example, ABa . Here A is called the initial

point and B, the terminal point or tip.

Magnitude or modulus of a is expressed as ABAB |||| a .

Note : The magnitude of a vector is always a non-negative real number.

Every vector AB has the following three characteristics:

Length : The length of AB will be denoted by || AB or AB.

Support : The line of unlimited length of which AB is a segment is called the support of the vector

AB .

Sense : The sense of AB is from A to B and that of BA is from B to A. Thus, the sense of a

directed line segment is from its initial point to the terminal point.

Types of Vector.

(1) Zero or null vector : A vector whose magnitude is zero is called zero or null vector and it is

represented by O .

The initial and terminal points of the directed line segment representing zero vector are coincident and its

direction is arbitrary.

B

A

a

(2) Unit vector : A vector whose modulus is unity, is called a unit vector. The unit vector in the direction

of a vector a is denoted by a , read as a cap. Thus, 1|| a .

a

a

of Magnitude

Vector

||

a

aa

Note : Unit vectors parallel to x-axis, y-axis and z-axis are denoted by i, j and k respectively.

Two unit vectors may not be equal unless they have the same direction.

(3) Like and unlike vectors : Vectors are said to be like when they have the same sense of direction

and unlike when they have opposite directions.

(4) Collinear or parallel vectors : Vectors having the same or parallel supports are called collinear

vectors.

(5) Co-initial vectors : Vectors having the same initial point are called co-initial vectors.

(6) Co-planar vectors : A system of vectors is said to be coplanar, if their supports are parallel to the same

plane.

Note : Two vectors having the same initial point are always coplanar but such three or more vectors

may or may not be coplanar.

(7) Coterminous vectors : Vectors having the same terminal point are called coterminous vectors.

(8) Negative of a vector : The vector which has the same magnitude as the vector a but opposite

direction, is called the negative of a and is denoted by a . Thus, if aPQ , then aQP .

(9) Reciprocal of a vector : A vector having the same direction as that of a given vector a but magnitude

equal to the reciprocal of the given vector is known as the reciprocal of a and is denoted by 1a . Thus, if

aa

1||,|| 1aa

Note : A unit vector is self reciprocal.

(10) Localized and free vectors : A vector which is drawn parallel to a given vector through a specified

point in space is called a localized vector. For example, a force acting on a rigid body is a localized vector as

its effect depends on the line of action of the force. If the value of a vector depends only on its length and

direction and is independent of its position in the space, it is called a free vector.

(11) Position vectors : The vector OA which represents the position of the point A with respect to a fixed point

O (called origin) is called position vector of the point A. If (x, y, z) are co-ordinates of the point A, then

kji zyxOA .

(12) Equality of vectors : Two vectors a and b are said to be equal, if

(i) |||| ba (ii) They have the same or parallel support and (iii) The same sense.

Rectangular resolution of a Vector in Two and Three dimensional systems.

(1) Any vector r can be expressed as a linear combination of two unit vectors i and j at right angle i.e.,

jir yx

The vector ix and jy are called the perpendicular component vectors

of r . The scalars x and y are called the components or resolved parts of r in

the directions of x-axis and y-axis respectively and the ordered pair (x, y) is

known as co-ordinates of point whose position vector is r .

Also the magnitude of 22 yxr and if be the inclination of r

with the x-axis, then )/(tan 1 xy

(2) If the coordinates of P are (x, y, z) then the position vector of r can be written as kjir zyx .

The vectors ji yx , and kz are called the right angled components of r .

The scalars zyx ,, are called the components or resolved parts of r in the directions of x-axis, y-axis and

z-axis respectively and ordered triplet (x, y, z) is known as coordinates of P whose position vector is r .

Also the magnitude or modulus of 222|| zyxrr

Direction cosines of r are the cosines of angles that the vector r

makes with the positive direction of x, y and z-axes.

||cos

222 r

x

zyx

xl ,

||cos

222 r

y

zyx

ym and

||cos

222 r

z

zyx

zn

Clearly, 1222 nml . Here POX , POY POZ and kji ,, are the unit vectors along

OZOYOX ,, respectively.

Example: 1 If a is a non-zero vector of modulus a and m is a non-zero scalar, then am is a unit vector if [MP PET 2002]

P(x, y)

X O i

j

Y

r

X

P(x, y, z)

r j

i k

Z

O

Y

(a) 1m (b) || am (c) ||

1

am (d) 2m

Solution: (c) As am is a unit vector, 1|| am 1|||| am ||

1||

am

||

1

am

Example: 2 For a non-zero vector a , the set of real numbers, satisfying |2||)5(| aax consists of all x such

that

(a) 30 x (b) 73 x (c) 37 x (d) 37 x

Solution: (b) We have, |2||)5(| aax

||2|||5| aax 2|5| x 252 x 73 x .

Example: 3 The direction cosines of the vector kji 543 are [Karnataka CET 2000]

(a) 5

1,

5

4,

5

3 (b)

2

1,

25

4,

25

3 (c)

2

1,

2

4,

2

3 (d)

2

1,

25

4,

25

3

Solution: (b) kjir 543 ; 255)4(3|| 222r

Hence, direction cosines are 25

5,

25

4,

25

3 i.e.,

2

1,

25

4,

25

3.

Properties of Vectors.

(1) Addition of vectors

(i) Triangle law of addition : If two vectors are represented by two

consecutive sides of a triangle then their sum is represented by the third side

of the triangle, but in opposite direction. This is known as the triangle law of

addition of vectors. Thus, if ba BCAB , and cAC then ACBCAB

i.e., cba .

(ii) Parallelogram law of addition : If two vectors are represented by

two adjacent sides of a parallelogram, then their sum is represented by the

diagonal of the parallelogram whose initial point is the same as the initial

point of the given vectors. This is known as parallelogram law of addition of

vectors.

Thus, if ba OBOA , and cOC

Then OCOBOA i.e., cba , where OC is a diagonal of the parallelogram OABC.

A B

C

a

b

c = a+b

c = a+b

b

a A

C B

O

(iii) Addition in component form : If the vectors are defined in terms of i , j and k, i.e., if

kjia 321 aaa and kjib 321 bbb , then their sum is defined as

kjiba )()()( 332211 bababa .

Properties of vector addition : Vector addition has the following properties.

(a) Binary operation : The sum of two vectors is always a vector.

(b) Commutativity : For any two vectors a and b , abba

(c) Associativity : For any three vectors ba, and c , cbacba )()(

(d) Identity : Zero vector is the identity for addition. For any vector a , 0aaa0

(e) Additive inverse : For every vector a its negative vector a exists such that 0aaaa )()(

i.e., )( a is the additive inverse of the vector a.

(2) Subtraction of vectors : If a and b are two vectors, then their subtraction ba is defined as

)( baba where b is the negative of b having magnitude equal to that of b and direction opposite to

b .

If kjia 321 aaa and kjib 321 bbb

Then kjiba )()()( 332211 bababa .

Properties of vector subtraction

(i) abba (ii) )()( cbacba

(iii) Since any one side of a triangle is less than the sum and greater than the difference of the other two

sides, so for any two vectors a and b, we have

(a) |||||| baba (b) |||||| baba

(c) |||||| baba (d) |||||| baba

(3) Multiplication of a vector by a scalar : If a is a vector and m is a scalar (i.e., a real number) then

am is a vector whose magnitude is m times that of a and whose direction is the same as that of a , if m is

positive and opposite to that of a , if m is negative.

Magnitude of || aa mm m (magnitude of a ) = || am

Again if kjia 321 aaa then kjia )()()( 321 mamamam

Properties of Multiplication of vectors by a scalar : The following are properties of multiplication of

vectors by scalars, for vectors ba, and scalars m, n

a

a + b

a+(b)= a b

b

b A 0

B

B

(i) )()()( aaa mmm (ii) aa mm )()(

(iii) )()()( aaa mnmnnm (iv) aaa nmnm )(

(v) baba mmm )(

(4) Resultant of two forces

QPR

cos2|| 22 PQQPRR

where || P = P, QQ || , cos

sintan

QP

Q

Deduction : When |||| QP , i.e., P = Q, 2

tancos1

sin

cos

sintan

PP

P;

2

Hence, the angular bisector of two unit vectors a and b is along the vector sum ba .

Important Tips

The internal bisector of the angle between any two vectors is along the vector sum of the corresponding

unit vectors.

The external bisector of the angle between two vectors is along the vector difference of the corresponding

unit vectors.

Position Vector .

If a point O is fixed as the origin in space (or plane) and P is any point, then OP

is called the position vector of P with respect to O.

P

Origin

O

r

(a) A

Internal

bisector

B(b)

External

bisector

O

)( ba )( ba

Q

R

P

If we say that P is the point r , then we mean that the position vector of P is r with respect to some origin

O.

(1) AB in terms of the position vectors of points A and B : If a and b are position vectors of points

A and B respectively. Then, ba OBOA ,

In OAB , we have abOAOBABOBABOA

AB = (Position vector of B) (Position vector of A)

AB = (Position vector of head) (Position vector of tail)

(2) Position vector of a dividing point

(i) Internal division : Let A and B be two points with position vectors a and b respectively, and let C be

a point dividing AB internally in the ratio m : n.

Then the position vector of C is given by

nm

nmOC

ab

(ii) External division : Let A and B be two points with position vectors a and b respectively and let C

be a point dividing AB externally in the ratio m : n.

Then the position vector of C is given by

nm

nmOC

ab

Important Tips

Position vector of the mid point of AB is 2

ba

If cb,a, are position vectors of vertices of a triangle, then position vector of its centroid is 3

cba

If dc,b,a, are position vectors of vertices of a tetrahedron, then position vector of its centroid is 4

dcba.

a

b

O

B

A

a

b

A

c O

C

m

n

B

A

a

b

c

B

C

O

Linear Combination of Vectors.

A vector r is said to be a linear combination of vectors .....,, cba etc, if there exist scalars x, y, z etc.,

such that ....cbar zyx

Examples : Vectors cbarcbar 23,32 21 are linear combinations of the vectors cba ,, .

(1) Collinear and Non-collinear vectors : Let a and b be two collinear vectors and let x be the unit

vector in the direction of a . Then the unit vector in the direction of b is x or x according as a and b are

like or unlike parallel vectors. Now, || aa x and xbb || .

xbb

aa ||

||

|| b

b

aa

||

|| ba , where

||

||

b

a.

Thus, if ba, are collinear vectors, then ba or ab for some scalar .

(2) Relation between two parallel vectors

(i) If a and b be two parallel vectors, then there exists a scalar k such that ba k .

i.e., there exist two non-zero scalar quantities x and y so that 0ba yx .

If a and b be two non-zero, non-parallel vectors then 0ba yx 0x and 0y .

Obviously 0ba yx

ba

0b0a

||

or

00,

or

,

yx

(ii) If kjia 321 aaa and kjib 321 bbb then from the property of parallel vectors, we have

3

3

2

2

1

1||b

a

b

a

b

aba

(3) Test of collinearity of three points : Three points with position vectors cb,a, are collinear iff there

exist scalars x, y, z not all zero such that 0cba zyx , where 0zyx . If jia 21 aa , jib 21 bb

and jic 21 cc , then the points with position vector cba ,, will be collinear iff 0

1

1

1

21

21

21

cc

bb

aa

.

(4) Test of coplanarity of three vectors : Let a and b two given non-zero non-collinear vectors. Then

any vectors r coplanar with a and b can be uniquely expressed as bar yx for some scalars x and y.

(5) Test of coplanarity of Four points : Four points with position vectors dcba ,,, are coplanar iff there

exist scalars x, y, z, u not all zero such that 0dcba uzyx , where 0uzyx .

Four points with position vectors

kjia 321 aaa , kjib 321 bbb , kjic 321 ccc , kjid 321 ddd

will be coplanar, iff 0

1

1

1

1

321

321

321

321

ddd

ccc

bbb

aaa

Linear Independence and Dependence of Vectors.

(1) Linearly independent vectors : A set of non-zero vectors naaa ,....., 21 is said to be linearly

independent, if 0.......... 212211 nnn xxxxxx 0aaa .

(2) Linearly dependent vectors : A set of vectors naaa ,....., 21 is said to be linearly dependent if there

exist scalars nxxx ,......,, 21 not all zero such that 0aaa nnxxx .........2211

Three vectors kjia 321 aaa , kjib 321 bbb and kjic 321 ccc will be linearly dependent

vectors iff 0

321

321

321

ccc

bbb

aaa

.

Properties of linearly independent and dependent vectors

(i) Two non-zero, non-collinear vectors are linearly independent.

(ii) Any two collinear vectors are linearly dependent.

(iii) Any three non-coplanar vectors are linearly independent.

(iv) Any three coplanar vectors are linearly dependent.

(v) Any four vectors in 3-dimensional space are linearly dependent.

Product of Two Vectors.

Product of two vectors is processed by two methods. When the product of two vectors results is a scalar

quantity, then it is called scalar product. It is also known as dot product because we are putting a dot (.) between

two vectors.

When the product of two vectors results is a vector quantity then this

product is called vector product. It is also known as cross product because

we are putting a cross () between two vectors.

(1) Scalar or Dot product of two vectors : If a and b are two non-

zero vectors and be the angle between them, then their scalar product (or

dot product) is denoted by ba . and is defined as the scalar cos|||| ba ,

where || and || ba are modulii of a and b respectively and 0 .

B

b

0 a A

Important Tips

Rba .

||||. baba

0.ba angle between a and b is acute.

0.ba angle between a and b is obtuse.

The dot product of a zero and non-zero vector is a scalar zero.

(i) Geometrical Interpretation of scalar product : Let a and b be

two vectors represented by OA and OB respectively. Let be the angle

between OA and OB . Draw OABL and OBAM .

From s OBL and OAM , we have cosOBOL and cosOAOM .

Here OL and OM are known as projection of b on a and a on b

respectively.

Now cos||||. baba = )cos(|| OBa = )(|| OLa

= )onof Projection()of Magnitude( aba .....(i)

Again, )cos|(|||cos||||. abbaba = )(||)cos(|| OMOA bb

a.b = (Magnitude of b) (Projection of a onb ) .....(ii)

Thus geometrically interpreted, the scalar product of two vectors is the product of modulus of either

vector and the projection of the other in its direction.

(ii) Angle between two vectors : If ba, be two vectors inclined at an angle , then, cos||||. baba

||||

.cos

ba

ba

||||

.cos 1

ba

ba

If kjia 321 aaa and kjib 321 bbb ; 23

22

21

23

22

21

3322111cosbbbaaa

bababa

(2) Properties of scalar product

(i) Commutativity : The scalar product of two vector is commutative i.e., abba .. .

(ii) Distributivity of scalar product over vector addition: The scalar product of vectors is distributive

over vector addition i.e.,

(a) cabacba ..)(. (Left distributivity) (b) acabacb ...)( (Right distributivity)

(iii) Let a and b be two non-zero vectors baba 0. .

B

b

O a A L

M

As kji ,, are mutually perpendicular unit vectors along the co-ordinate axes, therefore

0.. ijji ; ;0.. jkkj 0.. kiik .

(iv) For any vector 2||., aaaa .

As kji ,, are unit vectors along the co-ordinate axes, therefore 1||.2

iii , 1||. 2jjj and

1||. 2kkk

(v) If m is a scalar and ba, be any two vectors, then )(.).(.)( bababa mmm

(vi) If m, n are scalars and ba, be two vectors, then )(..)().(. babababa mnmnmnnm

(vii) For any vectors a and b , we have (a) bababa .)().()(. (b) baba .)(.)(

(viii) For any two vectors a and b , we have

(a) bababa .2|||||| 222 (b) bababa .2|||||| 222

(c) 22 ||||)(.)( bababa (d) |||||| baba ba||

(e) bababa 222 |||||| (f) bababa ||||

(3) Scalar product in terms of components.: If kjia 321 aaa and kjib 321 bbb ,

then, 332211. babababa . Thus, scalar product of two vectors is equal to the sum of the products of

their corresponding components. In particular, 2322

21

2||. aaaaaa .

Vector or Cross product of Two Vectors.

Let ba, be two non-zero, non-parallel vectors. Then the vector product

ba , in that order, is defined as a vector whose magnitude is |||| ba sin

where is the angle between a and b whose direction is perpendicular to

the plane of a and b in such a way that ba, and this direction constitute a

right handed system.

In other words, |||| baba sin where is the angle between a and b , is a unit vector

perpendicular to the plane of a and b such that ba ,, form a right handed system.

(1) Geometrical interpretation of vector product : If ba, be two non-zero, non-parallel vectors represented

by OA and OB respectively and let be the angle between them. Complete the parallelogram OACB. Draw

OABL .

In ,OBL OB

BLsin sin||sin bOBBL ......(i)

O

b

a

b

B C

A L O a

Now, baba sin|||| = ))(( BLOA

= Height) Base( = ) logramof paralle area( OACB

= Vector area of the parallelogram OACB

Thus, ba is a vector whose magnitude is equal to the area of the parallelogram having a and b as its

adjacent sides and whose direction is perpendicular to the plane of a and b such that ba ,, form a right

handed system. Hence ba represents the vector area of the parallelogram having adjacent sides along a and

b .

Thus, area of parallelogram OACB = || ba .

Also, area of 2

1OAB area of parallelogram OACB = ||

2

1||

2

1OBOAba

(2) Properties of vector product

(i) Vector product is not commutative i.e., if a and b are any two vectors, then abba , however,

)( abba

(ii) If ba, are two vectors and m is a scalar, then bababa mmm )(

(iii) If ba, are two vectors and m, n are scalars, then )()()( babababa mnnmmnnm

(iv) Distributivity of vector product over vector addition.

Let cba ,, be any three vectors. Then

(a) cabacba )( (Left distributivity)

(b) acabacb )( (Right distributivity)

(v) For any three vectors cba ,, we have cabacba )(

(vi) The vector product of two non-zero vectors is zero vector iff they are parallel (Collinear) i.e.,

baba0ba ,,|| are non-zero vectors.

It follows from the above property that 0aa for every non-zero vector a , which in turn implies that

0kkjjii

(vii) Vector product of orthonormal triad of unit vectors i, j, k using the definition of the vector product, we

obtain jikikjkji ,, , jkiijkkij ,,

(viii) Lagrange's identity: If a, b are any two vector then 2222 ).(|||||| bababa or

2222 ||||).(|| bababa

(3) Vector product in terms of components : If kjia 321 aaa and kjib 321 bbb .

Then, kjiba )()()( 122113312332 babababababa

321

321

bbb

aaa

kji

.

(4) Angle between two vectors : If is the angle between a and b , then ||||

||sin

ba

ba

Expression for sin : If kjia 321 aaa , kjib 321 bbb and be angle between a and b , then

))((

)()()(sin

23

22

21

23

22

21

21221

21331

223322

bbbaaa

babababababa

(5) (i) Right handed system of vectors : Three mutually perpendicular vectors cba ,, form a right

handed system of vector iff ,cba acb , bac

Example: The unit vectors ji , , k form a right-handed system,

jikikjkji ,,

(ii) Left handed system of vectors : The vectors cba ,, , mutually perpendicular to one another form a

left handed system of vector iff

cabbcaabc ,,

b

c a

k i

j

Y

X

Z

b

c

a

(6) Vector normal to the plane of two given vectors : If ba, be two non-zero, nonparallel vectors and

let be the angle between them. baba sin|||| where is a unit vector to the plane of a and b

such that ba ,, from a right-handed system.

baba ||)( ||

ba

ba

Thus, || ba

ba is a unit vector to the plane of a and b . Note that

|| ba

ba is also a unit vector

to the plane of a and b . Vectors of magnitude '' normal to the plane of a and b are given by

||

)(

ba

ba.

Moment of a Force and Couple.

(1) Moment of a force

(i) About a point : Let a force F be applied at a point P of a rigid

body. Then the moment of F about a point O measures the tendency of F

to turn the body about point O. If this tendency of rotation about O is in

anticlockwise direction, the moment is positive, otherwise it is negative.

Let r be the position vector of P relative to O. Then the moment or

torque of F about the point O is defined as the vector FM r .

If several forces are acting through the same point P, then the vector sum of the moment of the separate

forces about O is equal to the moment of their resultant force about O.

(ii) About a line: The moment of a force F acting at a point P about a line L is a scalar given by ar .)( F

where a is a unit vector in the direction of the line, and rOP , where O is any point on the line.

Thus, the moment of a force F about a line is the resolved part (component) along this line, of the

moment of F about any point on the line.

Note : The moment of a force about a point is a vector while the moment about a straight line is a scalar

quantity.

(2) Moment of a couple : A system consisting of a pair of equal

unlike parallel forces is called a couple. The vector sum of two forces of

a couple is always zero vector.

r

H

O

F

Q

P

r F

A N

r

F

F B

O

The moment of a couple is a vector perpendicular to the plane of couple and its magnitude is the product of

the magnitude of either force with the perpendicular distance between the lines of the forces.

FM r , where BAr

|||| FBAM = sin|||| BAF , where is the angle between BA and F

= )(|| BNF aF ||

where BNa is the arm of the couple and +ve or ve sign is to be taken according as the forces

indicate a counter-clockwise rotation or clockwise rotation.

(3) Rotation about an axis : When a rigid body rotates about a fixed axis ON with an angular velocity

, then the velocity v of a particle P is given by rv , where OPr and || (unit vector along ON)

Scalar Triple Product .

If cba ,, are three vectors, then their scalar triple product is defined as the dot product of two vectors a and

cb . It is generally denoted by a . )( cb or ][abc . It is read as box product of cba ,, . Similarly other scalar triple

products can be defined as bacacb .)(,.)( . By the property of scalar product of two vectors we can say,

cbacba .)()(.

(1) Geometrical interpretation of scalar triple product : The scalar triple product of three vectors is

equal to the volume of the parallelopiped whose three coterminous edges are represented by the given

vectors. cba ,, form a right handed system of vectors. Therefore ][.)( abccba = volume of the

parallelopiped, whose coterminous edges are a, b and c.

(2) Properties of scalar triple product

(i) If cba ,, are cyclically permuted, the value of scalar triple product remains the same. i.e.,

bacacbcba .)(.)(.)( or ][][][ bacacbcba

(ii) The change of cyclic order of vectors in scalar triple product changes the sign of the scalar triple

product but not the magnitude i.e., ][][][][ bcaabccabcba

N

P

r

O

(iii) In scalar triple product the positions of dot and cross can be interchanged provided that the cyclic

order of the vectors remains same i.e., )(..)( cbacba

(iv) The scalar triple product of three vectors is zero if any two of them are equal.

(v) For any three vectors cba ,, and scalar , ][][ cbacba

(vi) The scalar triple product of three vectors is zero if any two of them are parallel or collinear.

(vii) If dcba ,,, are four vectors, then ]][])[( [bcddcadcba

(viii) The necessary and sufficient condition for three non-zero non-collinear vectors cba ,, to be coplanar

is that 0][ cba i.e., cba ,, are coplanar 0][ bca .

(ix) Four points with position vectors a, b, c and d will be coplanar, if ][][][ badacdcba

][ cba .

(3) Scalar triple product in terms of components

(i) If kjia 321 aaa , kjib 321 bbb and kjic 321 ccc be three vectors.

Then,

333

222

111

][

cba

cba

cba

cba

(ii) If nmlbnmla 321321 , bbbaaa and nmlc 321 ccc , then ][][

321

321

321

nmlcba

ccc

bbb

aaa

(iii) For any three vectors ba, and c

(a) ][2][ cbaaccbba (b) 0][ accbba (c) 2][][ cbaaccbba

(4) Tetrahedron : A tetrahedron is a three-dimensional figure formed by

four triangle OABC is a tetrahedron with ABC as the base.

BCABOCOBOA ,,,, and CA are known as edges of the tetrahedron.

CAOBBCOA ,;, and ABOC, are known as the pairs of opposite edges. A

tetrahedron in which all edges are equal, is called a regular tetrahedron.

Properties of tetrahedron

(i) If two pairs of opposite edges of a tetrahedron are perpendicular, then the opposite edges of the third

pair are also perpendicular to each other.

(ii) In a tetrahedron, the sum of the squares of two opposite edges is the same for each pair.

(iii) Any two opposite edges in a regular tetrahedron are perpendicular.

Volume of a tetrahedron

a

b c

O

B(b) C(c)

A(a)

(i) The volume of a tetrahedron = altitude)ding (correspon base)of the area(3

1

= ||||2

1.

3

1EDACAB = EDACABEDACAB o ||for 0cos||||

6

1

= ][6

1][

6

1.)(

6

1ADACABADEAACABEDACAB .

Because EAACAB ,, are coplanar, so 0][ EAACAB

(ii) If cba ,, are position vectors of vertices A, B and C with respect to O, then volume of tetrahedron

OABC = ][6

1cba

(iii) If dcba ,,, are position vectors of vertices A, B, C, D of a tetrahedron ABCD, then

its volume = ][6

1adacab

(5) Reciprocal system of vectors : Let cba ,, be three non-coplanar vectors, and let

][,

][,

][ abc

bac

abc

acb

abc

cba .

c,b,a are said to form a reciprocal system of vectors for the vectors cba ,, .

If cba ,, and cba ,, form a reciprocal system of vectors, then

(i) 1... ccbbaa (ii) 0..;0..;0.. bcacabcbcaba

(iii) ][

1][

cbacba (v) cba ,, are non-coplanar iff so are cba ,, .

Vector Triple Product.

Let cba ,, be any three vectors, then the vectors )( cba and cba )( are called vector triple product of

cb,a, .

Thus, cbabcacba ).().()(

(1) Properties of vector triple product

(i) The vector triple product )( cba is a linear combination of those two vectors which are within

brackets.

(ii) The vector )( cbar is perpendicular to a and lies in the plane of b and c .

(iii) The formula cbabcacba ).().()( is true only when the vector outside the bracket is on the left

most side. If it is not, we first shift on left by using the properties of cross product and then apply the same

formula.

Thus, )}({)( cbaacb = }).().{( cbabca = bcacba ).().(

(iv) If kjia 321 aaa , kjib 321 bbb and kjic 321 ccc

then

122131132332

321)(

cbcbcbcbcbcb

aaa

kji

cba

Note : Vector triple product is a vector quantity.

cbacba )()(

Scalar product of Four Vectors.

)(.)( dcba is a scalar product of four vectors. It is the dot product of the vectors ba and dc .

It is a scalar triple product of the vectors ba, and dc as well as scalar triple product of the vectors cba ,

and d .

dbcb

dacadcba

..

..)(.)(

Vector product of Four Vectors.

(1) )()( dcba is a vector product of four vectors.

It is the cross product of the vectors ba and dc .

(2) dcbadcba }){()},({ are also different vector products of four vectors cba ,, and d .

Vector Equations.

Solving a vector equation means determining an unknown vector or a number of vectors satisfying the

given conditions. Generally, to solve a vector equation, we express the unknown vector as a linear combination

of three known non-coplanar vectors and then we determine the coefficients from the given conditions.

If ba, are two known non-collinear vectors, then baba ,, are three non-coplanar vectors.

Thus, any vector )( babar zyx where zyx ,, are unknown scalars.